ON MONOCHROMATIC LINEAR RECURRENCE SEQUENCES

In this paper we prove some van der Waerden type theorems for linear recurrence sequences. Under the assumption a(i-1) <= a(i)a(s-1) (i = 2, . . . , s), we extend results of G. Nyul and B. Rauf for sequences satisfying x(i) = a(1)x(i-s) + . . . + a(s)x(i-1) (i >= s + 1), where a(1), . . . , a(s) are positive integers. Moreover, we solve completely the same problem for sequences satisfying the binary recurrence relation x(i) = ax(i-1) - bx(i-2) (i >= 3) and x(1) < x(2), where a, b are positive integers with a >= b vertical bar 1.